Edge surface supercurrent and transport anomalies in Hall effect measurements in Nb

We report an experimental study of the edge surface superconductivity and the Hall effect in a Nb thin film Hall device. We show that the previously reported anomalous sign reversal of the Hall resistance across the superconducting-normal transition is due to the edge supercurrent. Large anomalous oscillations between the transverse voltage probes are observed in the field range between Hc2 and Hc3, in both the anti-symmetric (Hall) and symmetric (resistive) channels. These oscillations are extremely sensitive to a small tilting of the edge surface relative to the external magnetic field. We suggest that these oscillations may arise from the Aharonov-Bohm interference of the edge supercurrents. Introduction Saint-James and de Gennes (SJdG) showed1 that, within the Ginzburg-Landau theory of superconductivity2, there exists the possibility of a novel phenomenon of surface superconductivity, that with decreasing magnetic field, at a constant temperature below the superconducting transition temperature Tc, the superconducting order parameter ψ becomes non-zero first at the surface, at Hc3(T ), while the bulk of the sample remains in the normal state until the upper (for the bulk) critical field Hc2(T ) is reached. Early studies of surface superconductivity were on bulk samples with a flat surface so that one can easily align the sample surface with the magnetic field3, 4. The nature of the vortex matter physics in the bulk between Hc2(T ) and Hc3(T ) has been a subject of interest5, 6. Surprisingly, it was discovered that surface superconductivity can be found in thin films when the magnetic field is applied perpendicular to the film7, 8. On the other hand, if we go back to the original SJdG argument that the conditions for surface superconductivity are linearized Ginzburg-Landau equation with a simple boundary condition that the Cooper pairs cannot flow out of the sample: [(−ih̄ ∂ ∂x − 2eAx c )ψ]x=0 = 0 (1) where Ax is the vector potential of the magnetic field along the x-axis, the magnetic field H is along the z-axis. In the original SJdG model1, the boundary condition was set at x = 0, the y− z plane, and the system of interest is in x < 0. In fact, the SJdG theory, eq.(1), still works if the boundary is simply a thin film with a straight edge. For example, the film is in the x− y half plane, x < 0, the magnetic field is along the z-axis, the film edge is at x = 0, x > 0 is vacuum. Rightfully, Werner et al.7 called their effect the edge superconductivity. It was suggested8 that much of the broadening in the resistive transition from normal state to superconducting state in a magnetic field is due to the edge superconductivity. This striking conclusion calls for a careful re-examination of the magneto-transport measurements of thin-film superconductors. Recent theoretical efforts9, 10 have extended beyond the SJdG limit, to include surface roughness and disorder. With these motivations, we report a transport experiment on Nb thin films to investigate the longitudinal resistive transition and the Hall effect, two most important transport quantities associated with the superconducting-normal transition. The Hall effect in the superconducting-normal transition in a magnetic field has been a controversial subject11–21 with regard to the sign of the Hall coefficient and how it may change as a function of magnetic field or temperature. Our results suggest that the well-known anomaly of sign reversal in the Hall effect may in fact be caused by the edge superconductivity. Results and discussion In our experiment, Nb thin films are used such that our findings can be directly compared with previously reported results8, 14, 20. Fig.1 shows the representative SEM micrographs of a finished Hall bar device. As shown in Fig. 1(a), the dimensions of Hall bar region are 1 mm in length, 10 μm in width, and the voltage probes have a width of 5 μm. The films are of densely packed crystalline grains of sizes less than 50 nm (Fig.1(d)), consistent with other polycrystalline Nb films22. Fig.2 shows our main results of the resistive transition as a function of magnetic field H in our Nb sample, at T = 4.0 K, and at three different angles, as shown in the inset of Fig.2(a). The longitudinal resistance Rxx and transverse resistance Rxy are defined as Rxx =Vxx/Ix, Rxy= Vxy/Ix, respectively. Since the voltage signals due to the Hall effect change sign when the magnetic field direction is reversed, we plot both the symmetric and anti-symmetric parts of the Rxx and Rxy, as shown in Fig.2(a-d). Fig.2(a) shows the symmetric part of (Rxx, Rxx(H)+Rxx(−H))/2, which is the longitudinal resistance along the 1 mm long path. The transition from zero resistance superconducting state to the normal state is broad, over 4000 Oe. We defined Hc2 as the steepest slope in Rxx(H)vs.Hand Hc3 as where the longitudinal resistance first starts to decrease from the normal state value. Fig.2(b) shows the anti-symmetric part of Rxx, Rxx(H)−Rxx(−H))/2. This quantity should be zero if the sample geometry is perfect, i.e. the voltage probes are point-like. Instead, it exhibits peaks near Hc2, and slowly tapers off and vanishes at Hc3. This effect is especially pronounced when the sample is rotated by a small angle 10◦. The no-rotation configuration is when the magnetic field is perpendicular to the film (to within 1-2◦). The Hall resistance is the anti-symmetric part of Rxy, Rxy(H)−Rxy(−H))/2, as shown in Fig.2(c). When the magnetic field is perpendicular to the film surface, with decreasing field, the Hall resistance is a linear function of the magnetic field down to Hc2 where it drops to zero, with no sign change. However, when the sample is rotated by 10◦, either around x-axis or y-axis, (see Fig.1(a) inset for clarity), there are oscillatory behavior in the Hall resistance starting at Hc3. The Hall resistance clearly changes sign at Hc2. Similar oscillatory behavior in Rxy was reported in20. The symmetric part of the transverse resistance, shown in Fig.2(d), shows large oscillations as a function of magnetic field for all three field configurations. Later we will discuss the implications of these puzzling results. The measurement current in Fig.2 was 100 μA, the smallest current that still gives a low-noise measurement of the longitudinal and transverse resistance. Most of the features in Fig.2 remain for currents up to 500 μA. Fig.3 shows the effect of applied current in the Rxx and Rxy measurements. As shown in Fig.3(a), the broadened nature of the resistive transition is visible up to 500 μA, with shifts to features at Hc2 and Hc3. At 1 mA, the resistance drop at Hc2, shifted downward by about 1.5 kOe, is a large drop to zero resistance. The resistance tail due to the surface superconductivity is visible up to Hc3, the dashed line. The oscillatory and sign change behaviors in Fig.2(c,d) disappeared. In Fig.3(b), the large negative peak in the symmetric part of Rxy is still visible even at 1 mA. In Fig.3(c), the sign reversal effects in the Hall resistance is largely gone. Next, we wish to see if one can identify the bulk (interior) and the surface superconducting transitions in the temperature dependent measurements. In Fig.4(a), we converted the longitudinal resistance to resistivity ρxx and plotted vs. temperature. There are indeed clear signatures of two transitions. With decreasing temperature in a constant magnetic field, the sample resistivity first deviates from the normal state value (when zoom into the ρxx(T ) curve one can identify fluctuation effects) around Tc3(H). At Tc2(H), the resistivity drops to zero sharply. At I = 1 mA, it was clear in Fig.3 that there is a small but measurable effect in the transition field, this should result in about a change (downward) of 0.7 K in the Tc(H) determination. The measured Hc2 and Hc3 are summarized in Fig.4(b). As shown in Fig.4(b), the location of Tc2(H) or Hc3(T ) is much higher when measured at a smaller current. We suspect that the true Hc3(T ) line determined using infinitesimally small current is even higher. At 100μA, the Hc3(T ) intercepts the Hc2(T ) line. This is a real feature that was also observed in a Nb crystal with weak disorder5. It implies that the edge surface has a zero-field transition temperature lower than the zero-field transition temperature of the interior of the film. This is likely due to the ion damage during the IBE process, which is in fact visible in the close-up SEM micrograph in Fig.1(d). In summary of the experimental facts: (1) We confirmed that there is indeed an edge surface superconductivity effect in thin films of Nb in the resistive transition in both field and temperature dependent measurements of the longitudinal resistance, as found previously7, 8. (2) We found that the sign reversal phenomenon of the Hall effect is extremely sensitive to the orientation of the film device relative to the applied magnetic field. The sign reversal of the Hall coefficient appears only when the film is tilted relative to the magnetic field. When the film is perpendicular to the film, the sign-reversing Hall effect disappears. (3) When the film device is slightly tilted, large oscillations appear in the symmetric and anti-symmetric parts of the transverse resistance. Having established the connection between the anomalous sign reversal in the Hall coefficient, we wish to finally turn our attention to the oscillations in the anti-symmetric part of Rxx and the symmetric part of Rxy. Ideally, if the width of the voltage probes are reduced to zero, i.e. becoming a point, one should not see the anti-symmetric signals in Rxx, in Fig.2(b), and the symmetric signals in Rxy, in Fig.2(d). The fact that the voltage probes are of finite size, 5 μm in this experiment, we argue that these anomalous oscillation effects may well be the Aharonov-Bohm interference effect predicted theoretically9, 23. For clarity,


Introduction
Saint-James and de Gennes (SJdG) showed 1 that, within the Ginzburg-Landau theory of superconductivity 2 , there exists the possibility of a novel phenomenon of surface superconductivity, that with decreasing magnetic field, at a constant temperature below the superconducting transition temperature T c , the superconducting order parameter ψ becomes non-zero first at the surface, at H c3 (T ), while the bulk of the sample remains in the normal state until the upper (for the bulk) critical field H c2 (T ) is reached. Early studies of surface superconductivity were on bulk samples with a flat surface so that one can easily align the sample surface with the magnetic field 3,4 . The nature of the vortex matter physics in the bulk between H c2 (T ) and H c3 (T ) has been a subject of interest 5,6 .
Surprisingly, it was discovered that surface superconductivity can be found in thin films when the magnetic field is applied perpendicular to the film 7,8 . On the other hand, if we go back to the original SJdG argument that the conditions for surface superconductivity are linearized Ginzburg-Landau equation with a simple boundary condition that the Cooper pairs cannot flow out of the sample: where A x is the vector potential of the magnetic field along the x-axis, the magnetic field H is along the z-axis. In the original SJdG model 1 , the boundary condition was set at x = 0, the y − z plane, and the system of interest is in x < 0. In fact, the SJdG theory, eq.(1), still works if the boundary is simply a thin film with a straight edge. For example, the film is in the x − y half plane, x < 0, the magnetic field is along the z-axis, the film edge is at x = 0, x > 0 is vacuum. Rightfully, Werner et al. 7 called their effect the edge superconductivity. It was suggested 8 that much of the broadening in the resistive transition from normal state to superconducting state in a magnetic field is due to the edge superconductivity. This striking conclusion calls for a careful re-examination of the magneto-transport measurements of thin-film superconductors. Recent theoretical efforts 9, 10 have extended beyond the SJdG limit, to include surface roughness and disorder. With these motivations, we report a transport experiment on Nb thin films to investigate the longitudinal resistive transition and the Hall effect, two most important transport quantities associated with the superconducting-normal transition. The Hall effect in the superconducting-normal transition in a magnetic field has been a controversial subject [11][12][13][14][15][16][17][18][19][20][21] with regard to the sign of the Hall coefficient and how it may change as a function of magnetic field or temperature. Our results suggest that the well-known anomaly of sign reversal in the Hall effect may in fact be caused by the edge superconductivity.
In our experiment, Nb thin films are used such that our findings can be directly compared with previously reported results 8,14,20 . Fig.1 shows the representative SEM micrographs of a finished Hall bar device. As shown in Fig. 1(a), the dimensions of Hall bar region are 1 mm in length, 10 µm in width, and the voltage probes have a width of 5 µm. The films are of densely packed crystalline grains of sizes less than 50 nm ( Fig.1(d)), consistent with other polycrystalline Nb films 22 . Fig.2 shows our main results of the resistive transition as a function of magnetic field H in our Nb sample, at T = 4.0 K, and at three different angles, as shown in the inset of Fig.2(a). The longitudinal resistance R xx and transverse resistance R xy are defined as R xx = V xx /I x , R xy = V xy /I x , respectively. Since the voltage signals due to the Hall effect change sign when the magnetic field direction is reversed, we plot both the symmetric and anti-symmetric parts of the R xx and R xy , as shown in Fig.2(a-d). Fig.2(a) shows the symmetric part of (R xx , R xx (H) + R xx (−H))/2, which is the longitudinal resistance along the 1 mm long path. The transition from zero resistance superconducting state to the normal state is broad, over 4000 Oe. We defined H c2 as the steepest slope in R xx (H)vs.Hand H c3 as where the longitudinal resistance first starts to decrease from the normal state value. Fig.2(b) shows the anti-symmetric part of R xx , R xx (H) − R xx (−H))/2. This quantity should be zero if the sample geometry is perfect, i.e. the voltage probes are point-like. Instead, it exhibits peaks near H c2 , and slowly tapers off and vanishes at H c3 . This effect is especially pronounced when the sample is rotated by a small angle 10 • . The no-rotation configuration is when the magnetic field is perpendicular to the film (to within 1-2 • ).
The Hall resistance is the anti-symmetric part of R xy , R xy (H) − R xy (−H))/2, as shown in Fig.2(c). When the magnetic field is perpendicular to the film surface, with decreasing field, the Hall resistance is a linear function of the magnetic field down to H c2 where it drops to zero, with no sign change. However, when the sample is rotated by 10 • , either around x-axis or y-axis, (see Fig.1(a) inset for clarity), there are oscillatory behavior in the Hall resistance starting at H c3 . The Hall resistance clearly changes sign at H c2 . Similar oscillatory behavior in R xy was reported in 20 . The symmetric part of the transverse resistance, shown in Fig.2(d), shows large oscillations as a function of magnetic field for all three field configurations. Later we will discuss the implications of these puzzling results.
The measurement current in Fig.2 was 100 µA, the smallest current that still gives a low-noise measurement of the longitudinal and transverse resistance. Most of the features in Fig.2 remain for currents up to 500 µA. Fig.3 shows the effect of applied current in the R xx and R xy measurements. As shown in Fig.3(a), the broadened nature of the resistive transition is visible up to 500 µA, with shifts to features at H c2 and H c3 . At 1 mA, the resistance drop at H c 2, shifted downward by about 1.5 kOe, is a large drop to zero resistance. The resistance tail due to the surface superconductivity is visible up to H c3 , the dashed line. The oscillatory and sign change behaviors in Fig.2(c,d) disappeared. In Fig.3(b), the large negative peak in the symmetric part of R xy is still visible even at 1 mA. In Fig.3(c), the sign reversal effects in the Hall resistance is largely gone.
Next, we wish to see if one can identify the bulk (interior) and the surface superconducting transitions in the temperature dependent measurements. In Fig.4(a), we converted the longitudinal resistance to resistivity ρ xx and plotted vs. temperature. There are indeed clear signatures of two transitions. With decreasing temperature in a constant magnetic field, the sample resistivity first deviates from the normal state value (when zoom into the ρ xx (T ) curve one can identify fluctuation effects) around T c3 (H). At T c2 (H), the resistivity drops to zero sharply. At I = 1 mA, it was clear in Fig.3 that there is a small but measurable effect in the transition field, this should result in about a change (downward) of 0.7 K in the T c (H) determination. The measured H c2 and H c3 are summarized in Fig.4(b).
As shown in Fig.4(b), the location of T c2 (H) or H c3 (T ) is much higher when measured at a smaller current. We suspect that the true H c3 (T ) line determined using infinitesimally small current is even higher. At 100µA, the H c3 (T ) intercepts the H c2 (T ) line. This is a real feature that was also observed in a Nb crystal with weak disorder 5 . It implies that the edge surface has a zero-field transition temperature lower than the zero-field transition temperature of the interior of the film. This is likely due to the ion damage during the IBE process, which is in fact visible in the close-up SEM micrograph in Fig.1(d).
In summary of the experimental facts: (1) We confirmed that there is indeed an edge surface superconductivity effect in thin films of Nb in the resistive transition in both field and temperature dependent measurements of the longitudinal resistance, as found previously 7,8 . (2) We found that the sign reversal phenomenon of the Hall effect is extremely sensitive to the orientation of the film device relative to the applied magnetic field. The sign reversal of the Hall coefficient appears only when the film is tilted relative to the magnetic field. When the film is perpendicular to the film, the sign-reversing Hall effect disappears. (3) When the film device is slightly tilted, large oscillations appear in the symmetric and anti-symmetric parts of the transverse resistance.
Having established the connection between the anomalous sign reversal in the Hall coefficient, we wish to finally turn our attention to the oscillations in the anti-symmetric part of R xx and the symmetric part of R xy . Ideally, if the width of the voltage probes are reduced to zero, i.e. becoming a point, one should not see the anti-symmetric signals in R xx , in Fig.2(b), and the symmetric signals in R xy , in Fig.2(d). The fact that the voltage probes are of finite size, 5 µm in this experiment, we argue that these anomalous oscillation effects may well be the Aharonov-Bohm interference effect predicted theoretically 9,23 . For clarity, the anti-symmetric (Hall effect) and the symmetric (resistive) parts of R xy data in Fig.3 are replotted in an expanded view in Fig.5. As shown in Fig.5, with decreasing field, when H c3 is crossed, the anti-symmetric part of R xy shows the normal state Hall effect and drops to zero at H c2 , while the symmetric part of R xy exhibits a large increase at low currents 100-200 µA, this effect is suppressed at 500 µA, and a large negative peak at H c2 .
Here we propose a tentative model for how these anomalous transport properties may arise. Even though our Nb thin film is very thin, 120 nm in thickness, the IBE process creates sharp edge surfaces in the device. As illustrated in the inset of Fig.2(a), when the magnetic field is perpendicular to the film, all of these side surfaces are parallel to the magnetic field, creating a perfect condition for the SJdG mechanism 1 . With decreasing magnetic field, at H c3 (T ), the edge surface becomes superconducting first. As such, a diamagnetic supercurrent will appear at the edges, as shown by the dashed lines in Fig.5. In a transport measurement, when an external current is applied, there will be interplay between the applied current, and the spontaneously formed supercurrent. The latter has a topological feature similar to the edge currents in the integer quantum Hall effect [24][25][26] . Both are due to the orbital motion of the charged carriers coupled to the vector potential of the magnetic field (Landau orbits), they cannot flow out of the boundary, and are phase coherent (albeit for different physical mechanisms). The top and bottom panels in Fig.5 are for opposite field directions, the top for field into the page, and bottom for out of the page. The applied current is from left to right. When the magnetic field is reversed, the surface supercurrent (blue dashed lines) will reverse its direction (to remain diamagnetic). Since the voltage probes are fixed, A-B for the longitudinal voltage and B-C for the transverse voltage, the potential drops between A-B and B-C will depend on the field direction since the applied current and the edge supercurrent are additive in the top panel, but in subtraction in the bottom panel. When the applied current is small, it will also flow as Cooper pairs along the same paths as the diamagnetic screening supercurrent, namely the blue dashed lines and the red lines in Fig.5 are all on the edge surfaces, and they are all phase coherent and sensitive to the magnetic flux their closed paths enclose. Thus when the magnetic field is changed, the Aharonov-Bohm interference effects will occur between these currents. This current interference effect will manifest as resistance oscillations similar to multi-connected wire networks 27 . The novelty here is that the interference oscillations appear in a singly connected superconductor.

Sample preparation and characterization.
The Nb thin films used in this experiment have a thickness of 120 nm, deposited on SiO 2 /Si substrates by DC magnetron sputtering. The details of the deposition process can be found elsewhere 28,29 . Our Nb films do not show aging effect, there is no measurable change in the transition temperature over time (days). This is not surprising since aging effect was known to occur in Nb films with thickness at below 50 nm 30 .
The zero-field superconducting transition temperature T c of the Nb film was measured to be 8.85 K. The residual resistivity ratio (RRR) ρ xx (300 K)/ρ xx (10 K) is 4.3. The 10 K resistivity of our sample is 4.2 µΩcm. The carrier density n is measured to be 6.5×10 22 cm −3 according to the Hall resistance formula R xy = B/(nte) above T c where t is the film thickness (B = H in the CGS unit we adopt here). The Fermi wavevector is k F =(3π 2 n) 1/3 = 1.25 Å −1 .
It should be noted that the Hall coefficient of Nb is positive (hole-like) in the normal state due to its "jungle-gym" shaped Fermi surface [31][32][33][34][35][36] . This is what we found here as well. The Drude mean free path, l=¯h k F ne 2 ρ xx , of the carriers at 10 K is about 18.5 nm. The upper critical field H c2 at 4 K is 12.2 kOe. Using H c2 = φ o 2πξ 2 , we obtain a coherence length ξ = 16.3 nm. In comparison, the upper critical field H c2 (4K)=4.3 kOe was found for a Nb crystal with weak disorder 5 , giving a coherence length of 27.5 nm. For an annealed Nb crystal with a RRR=1550 31 , the upper critical field at 4.2 K was 2.6 kOe, giving a coherence length of 73 nm. Since ξ −1 = ξ o −1 + l −1 , where ξ o is the intrinisic coherence length, thus the coherence length in our Nb film is limited by the gain boundary scattering of the carriers.

Device fabrication and transport measurement .
Our Nb films were patterned into Hall bar devices using photolithography and an ion beam etching (IBE) process 37 . The IBE process utilizes the physical process of ion ballistic collision, and it results in sharp edges in the devices 37 .
The temperature dependent longitudinal resistances R xx and transverse resistances R xy of the Nb film device were measured by the standard four-probe method using the Physical Property Measurement System (PPMS-9, Quantum Design Inc.). The low resistance contacts were made using silver paste (Dupont 5025 Conductor Paste). The voltages were measured using AC resistance option within the PPMS, with alternating current directions to eliminate thermal electric signals. For Hall effect, the measurements were carried out in both +H and -H directions such that one can obtain both symmetric and anti-symmetric parts of the longitudinal and transverse resistances (or conductances). In the fixed T measurements, the sample temperature is regulated to within 20 mK of the set temperature. In the fixed H measurements, the field in the superconducting magnet remains constant.